Data Science Capstone Data Analysis
Kyla Mazzotta
2025-12-08
Data
- A dataset containing web scraped race results from the UTMB website
- Originally a json object, was then loaded before preprocessing
utmb_data <- fromJSON("utmb-race-data-raw.json", flatten=TRUE)Data Preprocessing
- Prior research shows that ultra marathons tend to be dominated by American and European athletes as there is little to no monetary incentive to perform well, a key factor differentiating marathon participants from ultra marathon participants.
- The main focus for this statistical analysis was therefore to isolate races that occurred in the United States to reduce potential variability in results across finish times with differen participant demographics country-wise.
- The UTMB race results were then filtered to only contain data on races located in the US
us_races <- utmb_data[
sapply(utmb_data, function(x) {
grepl("United States|USA|US$", x$`City / Country`, ignore.case = TRUE)
})
]- The dataset of US races was then converted into a data frame to prepare for exploratory data analysis (EDA)
- The original dataset contained information on the number of
participants in. each age 4-year age group. Since the there is little
variation in finish times with such small age groups, they were
consolidated into three bins:
- Under 35
- 35 - 59
- 60 and older
- Each age group bin was also adjusted to reoprt the percentage of athletes in each age group as opposed to number as the races vary in size and participant breakdowns.
us_df <- imap_dfr(us_races, ~{
results <- .x$Results
total_age <- sum(unlist(.x$Age))
tibble(
race_id = .y,
city_country = .x$`City / Country`,
date = .x$Date,
distance = .x$Distance,
elevation_gain = .x$`Elevation Gain`,
n_participants = length(results),
mean_time_hrs = mean(results, na.rm = TRUE),
median_time_hrs = median(results, na.rm = TRUE),
sd_time = sd(results, na.rm = TRUE),
min_time_hrs = min(results, na.rm = TRUE),
max_time_hrs = max(results, na.rm = TRUE),
pct_women = .x$Sex$Women / sum(unlist(.x$Sex)),
pct_age_u35 = sum(unlist(.x$Age[c("U18", "U20", "20-34")]), na.rm = TRUE) / total_age,
pct_age_35_59 = sum(unlist(.x$Age[c("35-39", "40-44", "45-49", "50-54", "55-59")]), na.rm = TRUE) / total_age,
pct_age_60_plus = sum(unlist(.x$Age[c("60-64", "65-69", "70-74", "75-79", "80+")]), na.rm = TRUE) / total_age
)
})- Since races also vary in number of participants, it was important to determine what the smallest acceptable race size would be to control for potential extreme race times.
- Total numbers if participants varied from 17 to 1842, therefore it was important to eliminate some of the smaller races from the data.
# min number of participants
min(us_df$n_participants, na.rm = TRUE)## [1] 1# 17
# max number of participants
max(us_df$n_participants, na.rm = TRUE)## [1] 1842# 1842
# distribution
summary(us_df$n_participants)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 1 48 91 132 164 1842- The 25th percentile was calculated to be races with 85 participants, therefore it was decided that only races with 85 or more participants would be analyzed as these races were sufficiently large enough while not eliminating too many data points
us_df <- us_df %>% filter(n_participants >= 85)- Distance and elevation variables were also converted from character strings to variables for analysis
colnames(us_df)[4] <- "distance_km"
colnames(us_df)[5] <- "elevation_gain_m"
us_df <- us_df %>%
mutate(
distance_km = parse_number(distance_km),
elevation_gain_m = parse_number(elevation_gain_m)
)will also only use distances greater than or equal to 50km
us_df <- us_df %>% filter(distance_km >= 50)center distance and elevation gain values
mean_distance_km <- mean(us_df$distance_km)
mean_elevation_m <- mean(us_df$elevation_gain_m)
us_df$distance_km <- us_df$distance_km - mean_distance_km
us_df$elevation_gain_m <- us_df$elevation_gain_m - mean_elevation_mreminder that now all distance and elevation values are centered
distance baseline = 85.0581
elevation baseline = 2564.2234
EDA
making a correlation matrix to see which factors should be included in regression remove pct_u35 to fix correlation issues (perfect multicollinearity)
correlation_matrix <- us_df %>% dplyr::select(distance_km, elevation_gain_m, pct_women, pct_age_35_59, pct_age_60_plus)
corrplot(cor(correlation_matrix))
Preliminary scatterplots for analysis
scatterplot of distance and elevation to see if linear relationship (maybe non-linear)
scatterplot indicates potential quadratic relationship between distance and mean finish time
ggplot(us_df, aes(x = distance_km, y = mean_time_hrs)) +
geom_point()
ggplot(us_df, aes(x = elevation_gain_m, y = mean_time_hrs)) +
geom_point()
Regression
prelim regression with just main effects remember that we are using pct_u35 as the base so it is not included in models (to avoid perfect multicollinearity)
prelim_regression <- lm(mean_time_hrs ~ distance_km + I(distance_km^2) + elevation_gain_m +
pct_women + pct_age_35_59 + pct_age_60_plus, data = us_df)
summary(prelim_regression)##
## Call:
## lm(formula = mean_time_hrs ~ distance_km + I(distance_km^2) +
## elevation_gain_m + pct_women + pct_age_35_59 + pct_age_60_plus,
## data = us_df)
##
## Residuals:
## Min 1Q Median 3Q Max
## -15.2897 -0.9539 0.0285 1.3338 14.8097
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.057e+01 7.952e-01 13.291 <2e-16 ***
## distance_km 1.054e-01 3.585e-03 29.394 <2e-16 ***
## I(distance_km^2) 7.398e-04 2.845e-05 26.001 <2e-16 ***
## elevation_gain_m 7.546e-04 7.126e-05 10.590 <2e-16 ***
## pct_women -2.082e-01 1.038e+00 -0.201 0.841
## pct_age_35_59 1.079e+00 1.075e+00 1.004 0.316
## pct_age_60_plus -2.523e+00 1.752e+00 -1.440 0.150
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.815 on 1188 degrees of freedom
## Multiple R-squared: 0.8983, Adjusted R-squared: 0.8978
## F-statistic: 1748 on 6 and 1188 DF, p-value: < 2.2e-16par(mfrow = c(1,2))
plot(prelim_regression, which=1, main="prelim model", pch=20)
plot(prelim_regression, which=2, main="prelim model", pch=20)
##### Using log regression as main model for robustness due to data not
appearing to be normally distributing, going to attempt logistic
regression mmodel
us_df <- us_df %>%
mutate(log_mean_time = log(mean_time_hrs))parsimonious model
log_regression <- lm(log_mean_time ~ distance_km + I(distance_km^2) + elevation_gain_m +
pct_women + pct_age_35_59 + pct_age_60_plus, data = us_df)
summary(log_regression)##
## Call:
## lm(formula = log_mean_time ~ distance_km + I(distance_km^2) +
## elevation_gain_m + pct_women + pct_age_35_59 + pct_age_60_plus,
## data = us_df)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.79419 -0.09046 0.01595 0.11109 0.50661
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2.402e+00 4.744e-02 50.645 <2e-16 ***
## distance_km 1.101e-02 2.138e-04 51.466 <2e-16 ***
## I(distance_km^2) -1.932e-05 1.697e-06 -11.383 <2e-16 ***
## elevation_gain_m 3.543e-05 4.251e-06 8.335 <2e-16 ***
## pct_women 7.844e-02 6.192e-02 1.267 0.205
## pct_age_35_59 -1.224e-03 6.414e-02 -0.019 0.985
## pct_age_60_plus -1.444e-01 1.045e-01 -1.382 0.167
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1679 on 1188 degrees of freedom
## Multiple R-squared: 0.889, Adjusted R-squared: 0.8884
## F-statistic: 1585 on 6 and 1188 DF, p-value: < 2.2e-16par(mfrow = c(1,2))
plot(log_regression, which=1, main="prelim model", pch=20)
plot(log_regression, which=2, main="prelim model", pch=20)
regressions with full model (interaction and higher order) adding only terms that are hypothesized to be significant and relevant
full_log_regression <- lm(log_mean_time ~ distance_km + I(distance_km^2) + elevation_gain_m +
pct_women + pct_age_35_59 + pct_age_60_plus +
distance_km*elevation_gain_m + distance_km*pct_women +
elevation_gain_m*pct_women, data = us_df)
summary(full_log_regression)##
## Call:
## lm(formula = log_mean_time ~ distance_km + I(distance_km^2) +
## elevation_gain_m + pct_women + pct_age_35_59 + pct_age_60_plus +
## distance_km * elevation_gain_m + distance_km * pct_women +
## elevation_gain_m * pct_women, data = us_df)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.78075 -0.08917 0.01458 0.11163 0.51964
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2.423e+00 4.805e-02 50.416 < 2e-16 ***
## distance_km 9.389e-03 6.351e-04 14.785 < 2e-16 ***
## I(distance_km^2) -1.490e-05 2.924e-06 -5.096 4.04e-07 ***
## elevation_gain_m 9.313e-05 1.518e-05 6.134 1.17e-09 ***
## pct_women 4.085e-02 6.409e-02 0.637 0.5240
## pct_age_35_59 -2.428e-02 6.440e-02 -0.377 0.7062
## pct_age_60_plus -1.748e-01 1.048e-01 -1.667 0.0957 .
## distance_km:elevation_gain_m -1.365e-07 8.572e-08 -1.592 0.1117
## distance_km:pct_women 5.189e-03 2.046e-03 2.537 0.0113 *
## elevation_gain_m:pct_women -2.043e-04 5.119e-05 -3.992 6.97e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.167 on 1185 degrees of freedom
## Multiple R-squared: 0.8905, Adjusted R-squared: 0.8896
## F-statistic: 1070 on 9 and 1185 DF, p-value: < 2.2e-16par(mfrow = c(1,2))
plot(full_log_regression, which=1, main="prelim model", pch=20)
plot(full_log_regression, which=2, main="prelim model", pch=20)
regression with reduced model after stepwise backward elimination
log_backward_step_model <- stepAIC(full_log_regression, direction = "backward")## Start: AIC=-4267.96
## log_mean_time ~ distance_km + I(distance_km^2) + elevation_gain_m +
## pct_women + pct_age_35_59 + pct_age_60_plus + distance_km *
## elevation_gain_m + distance_km * pct_women + elevation_gain_m *
## pct_women
##
## Df Sum of Sq RSS AIC
## - pct_age_35_59 1 0.00396 33.042 -4269.8
## <none> 33.038 -4268.0
## - distance_km:elevation_gain_m 1 0.07065 33.108 -4267.4
## - pct_age_60_plus 1 0.07750 33.115 -4267.2
## - distance_km:pct_women 1 0.17939 33.217 -4263.5
## - elevation_gain_m:pct_women 1 0.44421 33.482 -4254.0
## - I(distance_km^2) 1 0.72389 33.762 -4244.1
##
## Step: AIC=-4269.82
## log_mean_time ~ distance_km + I(distance_km^2) + elevation_gain_m +
## pct_women + pct_age_60_plus + distance_km:elevation_gain_m +
## distance_km:pct_women + elevation_gain_m:pct_women
##
## Df Sum of Sq RSS AIC
## <none> 33.042 -4269.8
## - distance_km:elevation_gain_m 1 0.07254 33.114 -4269.2
## - pct_age_60_plus 1 0.07357 33.115 -4269.2
## - distance_km:pct_women 1 0.17742 33.219 -4265.4
## - elevation_gain_m:pct_women 1 0.44025 33.482 -4256.0
## - I(distance_km^2) 1 0.71999 33.762 -4246.1summary(log_backward_step_model)##
## Call:
## lm(formula = log_mean_time ~ distance_km + I(distance_km^2) +
## elevation_gain_m + pct_women + pct_age_60_plus + distance_km:elevation_gain_m +
## distance_km:pct_women + elevation_gain_m:pct_women, data = us_df)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.78083 -0.09029 0.01419 0.11212 0.52128
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2.406e+00 1.873e-02 128.432 < 2e-16 ***
## distance_km 9.372e-03 6.331e-04 14.804 < 2e-16 ***
## I(distance_km^2) -1.481e-05 2.913e-06 -5.084 4.30e-07 ***
## elevation_gain_m 9.281e-05 1.515e-05 6.124 1.24e-09 ***
## pct_women 3.932e-02 6.394e-02 0.615 0.5387
## pct_age_60_plus -1.667e-01 1.026e-01 -1.625 0.1044
## distance_km:elevation_gain_m -1.381e-07 8.558e-08 -1.614 0.1069
## distance_km:pct_women 5.156e-03 2.043e-03 2.524 0.0117 *
## elevation_gain_m:pct_women -2.025e-04 5.094e-05 -3.975 7.46e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1669 on 1186 degrees of freedom
## Multiple R-squared: 0.8905, Adjusted R-squared: 0.8897
## F-statistic: 1205 on 8 and 1186 DF, p-value: < 2.2e-16par(mfrow = c(1,2))
plot(log_backward_step_model, which=1, main="backward", pch=20)
plot(log_backward_step_model, which=2, main="backward", pch=20)
regression using stepwise elimination (both directions)
log_both_step_model <- stepAIC(full_log_regression, direction = "both")## Start: AIC=-4267.96
## log_mean_time ~ distance_km + I(distance_km^2) + elevation_gain_m +
## pct_women + pct_age_35_59 + pct_age_60_plus + distance_km *
## elevation_gain_m + distance_km * pct_women + elevation_gain_m *
## pct_women
##
## Df Sum of Sq RSS AIC
## - pct_age_35_59 1 0.00396 33.042 -4269.8
## <none> 33.038 -4268.0
## - distance_km:elevation_gain_m 1 0.07065 33.108 -4267.4
## - pct_age_60_plus 1 0.07750 33.115 -4267.2
## - distance_km:pct_women 1 0.17939 33.217 -4263.5
## - elevation_gain_m:pct_women 1 0.44421 33.482 -4254.0
## - I(distance_km^2) 1 0.72389 33.762 -4244.1
##
## Step: AIC=-4269.82
## log_mean_time ~ distance_km + I(distance_km^2) + elevation_gain_m +
## pct_women + pct_age_60_plus + distance_km:elevation_gain_m +
## distance_km:pct_women + elevation_gain_m:pct_women
##
## Df Sum of Sq RSS AIC
## <none> 33.042 -4269.8
## - distance_km:elevation_gain_m 1 0.07254 33.114 -4269.2
## - pct_age_60_plus 1 0.07357 33.115 -4269.2
## + pct_age_35_59 1 0.00396 33.038 -4268.0
## - distance_km:pct_women 1 0.17742 33.219 -4265.4
## - elevation_gain_m:pct_women 1 0.44025 33.482 -4256.0
## - I(distance_km^2) 1 0.71999 33.762 -4246.1summary(log_both_step_model)##
## Call:
## lm(formula = log_mean_time ~ distance_km + I(distance_km^2) +
## elevation_gain_m + pct_women + pct_age_60_plus + distance_km:elevation_gain_m +
## distance_km:pct_women + elevation_gain_m:pct_women, data = us_df)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.78083 -0.09029 0.01419 0.11212 0.52128
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2.406e+00 1.873e-02 128.432 < 2e-16 ***
## distance_km 9.372e-03 6.331e-04 14.804 < 2e-16 ***
## I(distance_km^2) -1.481e-05 2.913e-06 -5.084 4.30e-07 ***
## elevation_gain_m 9.281e-05 1.515e-05 6.124 1.24e-09 ***
## pct_women 3.932e-02 6.394e-02 0.615 0.5387
## pct_age_60_plus -1.667e-01 1.026e-01 -1.625 0.1044
## distance_km:elevation_gain_m -1.381e-07 8.558e-08 -1.614 0.1069
## distance_km:pct_women 5.156e-03 2.043e-03 2.524 0.0117 *
## elevation_gain_m:pct_women -2.025e-04 5.094e-05 -3.975 7.46e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1669 on 1186 degrees of freedom
## Multiple R-squared: 0.8905, Adjusted R-squared: 0.8897
## F-statistic: 1205 on 8 and 1186 DF, p-value: < 2.2e-16par(mfrow = c(1,2))
plot(log_both_step_model, which=1, main="both directions", pch=20)
plot(log_both_step_model, which=2, main="both directions", pch=20)
checking VIF of log stepwise regression models
vif(log_backward_step_model, type = "predictor")## GVIFs computed for predictors## GVIF Df GVIF^(1/(2*Df))
## distance_km 1.110801 7 1.007534
## elevation_gain_m 1.110801 7 1.007534
## pct_women 1.110801 7 1.007534
## pct_age_60_plus 1.110801 1 1.053946
## Interacts With
## distance_km I(distance_km^2), elevation_gain_m, pct_women
## elevation_gain_m distance_km, pct_women
## pct_women distance_km, elevation_gain_m
## pct_age_60_plus --
## Other Predictors
## distance_km pct_age_60_plus
## elevation_gain_m pct_age_60_plus
## pct_women pct_age_60_plus
## pct_age_60_plus distance_km, elevation_gain_m, pct_womenlook for outliers using cooks distance
cooks_distance <- cooks.distance(log_both_step_model)
threshold <- 4 / nrow(us_df)
influential_observations <- which(cooks_distance > threshold)
plot(log_both_step_model)
rerun regression without outliers to see if conclusions change (sensitivity analysis)
remove outliers
keep_rows <- which(cooks_distance <= threshold)
us_df_no_outliers <- us_df[keep_rows, ]stepwise regression using data with outliers removes to check sensitivity and robustness
no_outlier_log_regression <- lm(log_mean_time ~ distance_km + I(distance_km^2) + elevation_gain_m +
pct_women + pct_age_35_59 + pct_age_60_plus +
distance_km*elevation_gain_m + distance_km*pct_women +
elevation_gain_m*pct_women, data = us_df_no_outliers)
outlierless_log_both_step_model <- stepAIC(no_outlier_log_regression, direction = "both")## Start: AIC=-4535.9
## log_mean_time ~ distance_km + I(distance_km^2) + elevation_gain_m +
## pct_women + pct_age_35_59 + pct_age_60_plus + distance_km *
## elevation_gain_m + distance_km * pct_women + elevation_gain_m *
## pct_women
##
## Df Sum of Sq RSS AIC
## - pct_age_60_plus 1 0.00024 19.344 -4537.9
## - pct_age_35_59 1 0.00255 19.346 -4537.7
## - distance_km:pct_women 1 0.00850 19.352 -4537.4
## <none> 19.343 -4535.9
## - elevation_gain_m:pct_women 1 0.13276 19.476 -4530.2
## - distance_km:elevation_gain_m 1 0.51286 19.856 -4508.5
## - I(distance_km^2) 1 0.92339 20.267 -4485.6
##
## Step: AIC=-4537.88
## log_mean_time ~ distance_km + I(distance_km^2) + elevation_gain_m +
## pct_women + pct_age_35_59 + distance_km:elevation_gain_m +
## distance_km:pct_women + elevation_gain_m:pct_women
##
## Df Sum of Sq RSS AIC
## - pct_age_35_59 1 0.00293 19.347 -4539.7
## - distance_km:pct_women 1 0.00830 19.352 -4539.4
## <none> 19.344 -4537.9
## + pct_age_60_plus 1 0.00024 19.343 -4535.9
## - elevation_gain_m:pct_women 1 0.13253 19.476 -4532.2
## - distance_km:elevation_gain_m 1 0.52069 19.864 -4510.1
## - I(distance_km^2) 1 0.92786 20.272 -4487.3
##
## Step: AIC=-4539.71
## log_mean_time ~ distance_km + I(distance_km^2) + elevation_gain_m +
## pct_women + distance_km:elevation_gain_m + distance_km:pct_women +
## elevation_gain_m:pct_women
##
## Df Sum of Sq RSS AIC
## - distance_km:pct_women 1 0.00858 19.355 -4541.2
## <none> 19.347 -4539.7
## + pct_age_35_59 1 0.00293 19.344 -4537.9
## + pct_age_60_plus 1 0.00062 19.346 -4537.7
## - elevation_gain_m:pct_women 1 0.13564 19.482 -4533.9
## - distance_km:elevation_gain_m 1 0.51896 19.866 -4512.0
## - I(distance_km^2) 1 0.93827 20.285 -4488.6
##
## Step: AIC=-4541.21
## log_mean_time ~ distance_km + I(distance_km^2) + elevation_gain_m +
## pct_women + distance_km:elevation_gain_m + elevation_gain_m:pct_women
##
## Df Sum of Sq RSS AIC
## <none> 19.355 -4541.2
## + distance_km:pct_women 1 0.00858 19.347 -4539.7
## + pct_age_35_59 1 0.00321 19.352 -4539.4
## + pct_age_60_plus 1 0.00026 19.355 -4539.2
## - elevation_gain_m:pct_women 1 0.16544 19.521 -4533.7
## - distance_km:elevation_gain_m 1 0.53072 19.886 -4512.9
## - I(distance_km^2) 1 0.95482 20.310 -4489.2summary(outlierless_log_both_step_model)##
## Call:
## lm(formula = log_mean_time ~ distance_km + I(distance_km^2) +
## elevation_gain_m + pct_women + distance_km:elevation_gain_m +
## elevation_gain_m:pct_women, data = us_df_no_outliers)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.65988 -0.07322 0.00263 0.08543 0.37731
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2.445e+00 1.634e-02 149.564 < 2e-16 ***
## distance_km 1.210e-02 2.015e-04 60.056 < 2e-16 ***
## I(distance_km^2) -2.503e-05 3.375e-06 -7.417 2.38e-13 ***
## elevation_gain_m 8.128e-05 1.181e-05 6.882 9.80e-12 ***
## pct_women 4.296e-02 5.676e-02 0.757 0.44929
## distance_km:elevation_gain_m -5.275e-07 9.540e-08 -5.529 4.00e-08 ***
## elevation_gain_m:pct_women -1.187e-04 3.846e-05 -3.087 0.00207 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1318 on 1115 degrees of freedom
## Multiple R-squared: 0.9225, Adjusted R-squared: 0.9221
## F-statistic: 2211 on 6 and 1115 DF, p-value: < 2.2e-16par(mfrow = c(1,2))
plot(outlierless_log_both_step_model, which=1, main="both directions", pch=20)
plot(outlierless_log_both_step_model, which=2, main="both directions", pch=20)
plot(log_both_step_model)
these clusters are due to the most common distances 50k, 50mi, 100k, and 100mi
deciding to use the outlier-free model
vif analysis
vif(outlierless_log_both_step_model, type = "predictor")## GVIFs computed for predictors## GVIF Df GVIF^(1/(2*Df))
## distance_km 14.117138 4 1.392254
## elevation_gain_m 1.000000 6 1.000000
## pct_women 4.406172 3 1.280394
## Interacts With Other Predictors
## distance_km I(distance_km^2), elevation_gain_m pct_women
## elevation_gain_m distance_km, pct_women --
## pct_women elevation_gain_m distance_kmmaking scatterplots to put on the poster to show distributions
scatterplot of distance and mean time (with outliers)
ggplot(us_df, aes(x = distance_km, y = mean_time_hrs)) +
geom_point(alpha = 0.6, size = 3) +
geom_smooth(method = "lm", se = FALSE, col = "#f1e60e") +
labs(
x = "Race Distance (km)",
y = "Mean Finish Time (hrs)",
title = "Race Distance Vs Mean Finish Time"
) +
theme_minimal(base_size = 15)## `geom_smooth()` using formula = 'y ~ x'
first scatterplot of mean time and distance with outliers removed
ggplot(us_df_no_outliers, aes(x = distance_km, y = mean_time_hrs)) +
geom_point(alpha = 0.6, size = 3) +
geom_smooth(method = "lm", se = FALSE, col = "#f1e60e") +
labs(
x = "Race Distance (km)",
y = "Mean Finish Time (hrs)",
title = "Race Distance Vs Mean Finish Time"
) +
theme_minimal(base_size = 15)## `geom_smooth()` using formula = 'y ~ x'
second scatterplot of mean time and elevation gain (with outliers)
ggplot(us_df, aes(x = elevation_gain_m, y = mean_time_hrs)) +
geom_point(alpha = 0.6, size = 3) +
geom_smooth(method = "lm", se = FALSE, col = "#f1e60e") +
labs(
x = "Elevation Gain (m)",
y = "Mean Finish Time (hrs)",
title = "Elevation Gain Vs Mean Finish Time"
) +
theme_minimal(base_size = 15)## `geom_smooth()` using formula = 'y ~ x'
second scatterplot of mean time and elevation gain with outliers removed
ggplot(us_df_no_outliers, aes(x = elevation_gain_m, y = mean_time_hrs)) +
geom_point(alpha = 0.6, size = 3) +
geom_smooth(method = "lm", se = FALSE, col = "#f1e60e") +
labs(
x = "Elevation Gain (m)",
y = "Mean Finish Time (hrs)",
title = "Elevation Gain Vs Mean Finish Time"
) +
theme_minimal(base_size = 15)## `geom_smooth()` using formula = 'y ~ x'